Trying to combine red, green and blue to produce white

What you are seeing at a distance is not black. It is a darkish shade of gray, RGB gray 85,85,85. The reason you aren't seeing "white" is because each of those three rectangles has an HSV value of only 33% and you are seeing that merged square against a white background.

That merged square will appear to be whitish if you make the background black rather than white and view the screen in a very dark setting.

While David Hamman's answer is correct, I wanted to expand a little bit on his answer:

When you use a CRT, you are looking at emitted light. In the case of emission, there is no "absolute" white - something will only look gray in comparison to something else with the same color ratio but brighter. When you turn up the brightness on your monitor, white is still white - just "whiter". In a dark room, even a CRT whose intensity is turned down a lot appears to render white as white.

By contrast when you have a sheet of paper with colors on it, what is really happening is that you have absorption of certain wavelengths / colors. A blue piece of paper absorbs everything except blue; ditto for red, and green. So what you have on your three rectangles is three pieces of paper that absorb 2/3 of the incident light:

      red  green  blue
red   100     0     0 
green   0   100     0
blue    0     0   100

When you illuminate the entire card with 100% white light, you get a red reflection from just 1/3 of the card - and the same for green and blue. If you have these three cards against a white background, the background will appear approximately 3x brighter as every part of the background reflects all three colors.

If you made the background gray (which is really just a "darker shade of white"), and shone a bright light onto the setup, you could actually persuade yourself that the background, and the three color sample, were white.